This invention relates to an area of logical processing of continuous signals. While regular Boolean logic deals with two discrete (crisp) signals 0 and 1, which are usually designate two (False and Truth) values, the continuous-valued logic operates on any value in closed interval between real numbers 0 and 1 including the interval's endpoints. Any continuous electrical signal can be normalized and brought to this interval, i.e. made a continuous-valued logical signal. Such signals are usually considered to be equivalent to probability, or degree of truth, or degree of participation. Since this kind of algebraic logic employs probabilistic disjunction, it is commonly referred to as probabilistic logic (Nilsson, N.J., 1986, Probabilistic logic, Artificial Intelligence 28(1): 71-87). The continuous-valued logic is not the same as multi-valued (or many-valued) logic. The difference is in the fact that usually the multi-valued logic is understood as a logic that deals with a countable but finite number of states: for example, three (in a case of ternary logic), or four, or as a radix of any other counting system. Otherwise, the continuous-valued logic is a logic, which has an indefinite number of states.
Continuous-valued logic is a kind of algebraic logic that uses algebraic formulas to express underlying logical operations. The simplest and effective enough set of three basic algebraic logical operations usually contains: product conjunction z=x AND y=xy, probabilistic disjunction z=x OR y=x+y−xy, and Godel's negation z=NOT x=1−x or z=NOT y=1−y. These operations can be implemented either in analog or in digital form.
A major problem associated with the logical processing of continuous signals is that they do not exactly follow the same rules as a regular Boolean logic, necessitating many special considerations to be taken into account, such as:                (a) Uncertainty—It is possible to define a large and indefinite number of continuous-valued logical functions that satisfy discrete Boolean truth tables (at the interval endpoints 0 and 1). Therefore, continuous logical functions are not standardized, not repeatable and not deterministic expressions.        (b) Speed—Despite its simplistic appearance, the use of probabilistic disjunction is associated with an increase of computational time which is especially significant when a large number of input variables involved.        (c) Compatibility—Continuous-valued logical expressions have to cover both continuous signals inside their domains and discrete signals at the endpoints of this interval as well.        (d) Ambiguity—What method to apply while generating the continuous logical functions. Although it is known that some such functions are polynomials, it is still questionable how to define these polynomials in order to lower their degree.        
A significant number of existing art solutions (U.S. Pat. Nos. 5,398,199, 6,133,754, and 7,355,444) relate to logical processing of multi-valued, but not continuous-valued signals. Many of existing art solutions (U.S. Pat. Nos. 5,463,565 and 5,463,572) while also being multi-valued are suitable to a single specific logical function like AND and OR but not for assessing any required functions. An effort has been made to propose universal logical processors (ALU) capable of generating various logical functions (U.S. Pat. Nos. 4,914,614 and 5,227,993), however these solutions also relate only to the multi-valued and not to the continuous-valued logic.
Mills (U.S. Pat. No. 5,770,966) shows diode and transistor analog logical arrays compounded of continuous Lukaciewicz implication and negated implication circuits. Such arrays model a human eye retina. Wang (U.S. Pat. No. 5,799,296) proposed a logical system using real-time neural network comprised of continuous variables that do not utilize membership functions and employ an approximation of a neuron with a multiplier. All these solutions are based on the use of simple basic logical operations (in order to improve level of integration) and are often realized as analog circuits (in order to increase speed). Gershenfeld (application for U.S. patent Ser. No. 12/422,491) proposed an array of analog circuits which incorporated statistical signal processing. In this solution, probabilities were assigned to certain combinations of two Boolean logical input variables (00, 01, 10, and 11) and used in the logical processing cell (i.e. probabilities, with which these combinations of discrete logical signals came, were processed).
Of a special interest is a logical processing unit capable of handling continuous-valued signals such as the above mentioned ALU for multi-valued logic. Such universal logical processor could be used in various applications that involve artificial intelligence, logical reasoning, neural networks, and fuzzy control. Murphy (U.S. Pat. No. 5,077,677) has proposed a probabilistic inference gate that processes probabilities of certain events (which are continuous signals) while treating them as logical entities. In this solution, any relationship has to be expressed in terms of probabilistic implications. Although effective in some specific situations, particularly in those, at which experts' reasoning inference is approximated, generally there are several drawbacks associated with this approach:                (a) Logical functions are not expressed in standard algebraic form;        (c) Such probabilistic implication functions are not always compatible with Boolean discrete logical functions at endpoints of logical interval;        (d) These functions disallow to be expressed as disjunctions of maxterms (in DNF) or conjunctions of maxterms (in CNF);        (e) Such defined logical functions do not follow De Morgan laws and therefore cannot be used in the functions' simplification process;        (f) Definitions of these multi-staged implications are increasingly complicated in cases of larger number of input variables.        
It is common practice to define the Boolean (discrete) logical functions while using DNF or CNF. However, it has been neither demonstrated that the use of the CNF and DNF in association with the continuous-valued logic brings important advantages nor shown how to define such functions this way. Theory establishing relationships among such continuous-valued objects as direct and negated logical functions, their minterms and maxterms, and continuous input signals is in process of development.
In the prior art, no standard method for deterministic definition of continuous logical functions was proposed. Existing art solutions disallow of construction of universal logical processor capable of generating arbitrary continuous-valued logical functions. Also, proposed earlier solutions are often not uniquely defined and not guarantee repeatable results in some circumstances. Usually, the existing art solutions are not optimized for performance (not guarantee the minimum execution time). As a result, none of the existing art solutions can be characterized by a set of features to be considered a standardized continuous-valued logic, even though they might meet the requirements of some applications.
It is, therefore, an object of the invention to guarantee compatibility between continuous and discrete logical functions.
It is an object of the invention as well to provide a universal logical processor to handle any practical arbitrary number of continuous input variables and deliver arbitrary continuous logical function.
It is also an object of the invention to provide a logical processor capable of defining the continuous logical functions with complete determinism while guaranteeing minimum processing time.
It is further an object of the invention as well to provide a continuous-valued logical processor that adheres to CNF and DNF rules similar to regular Boolean logic.